Problem: A group of adults and kids went to see a movie. Tickets cost $$5.00$ each for adults and $$3.50$ each for kids, and the group paid $$41.50$ in total. There were $7$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Solution: Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${5x+3.5y = 41.5}$ ${x = y-7}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-7}$ for $x$ in the first equation. ${5}{(y-7)}{+ 3.5y = 41.5}$ Simplify and solve for $y$ $ 5y-35 + 3.5y = 41.5 $ $ 8.5y-35 = 41.5 $ $ 8.5y = 76.5 $ $ y = \dfrac{76.5}{8.5} $ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into ${x = y-7}$ to find $x$ ${x = }{(9)}{ - 7}$ ${x = 2}$ You can also plug ${y = 9}$ into ${5x+3.5y = 41.5}$ and get the same answer for $x$ ${5x + 3.5}{(9)}{= 41.5}$ ${x = 2}$ There were $2$ adults and $9$ kids.